#4452690

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,6,5)(3,8,4,7)(9,10)(11,14)(12,15)(13,16)(18,20)(19,24)(25,26,30,29)(27,32,28,31);(1,3,6,4)(2,7,5,8)(9,11)(10,12)(13,15)(14,16)(17,18,21,20)(19,23,24,22)(25,26,30,29)(27,31,28,32);(9,11)(10,12)(13,15)(14,16)(17,19)(18,22)(20,23)(21,24)(25,27,30,28)(26,31,29,32);(1,4,6,3)(2,7,5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)(25,28,30,27)(26,32,29,31);(1,5,6,2)(3,8,4,7)(9,12,13,14)(10,15,16,11)(17,20,21,18)(19,23,24,22)(25,29,30,26)(27,32,28,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=hi,d^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4359886);